Chapter 14 - Internal Balancing and Model Reduction

Abstract

This chapter presents internal balancing, model reduction via internal balancing and Schur decomposition, and Hankel-norm approximation. The problem of constructing a reduced-order model such that the Hankel-norm error is minimized is called an Optimal Hankel-norm approximation problem. A widely used practice of model reduction is to first find a balanced realization and then to truncate the balanced realization in an appropriate manner to obtain a reduced order model. The chapter discusses the balancing of a continuous-time system, where two algorithms are described. The first algorithm constructs internal balancing of a stable, controllable, and observable system, whereas the second algorithm is designed to extract a balanced realization, if the original system is not minimal. The chapter presents that a reduced order model constructed by truncating a balanced realization remains stable and the H∞-norm error is bounded. The chapter shows that the transfer function matrix of the reduced-order model obtained by the Schur method is the same as that of the model reduction procedure via internal balancing. The advantages of the Schur and the square-root methods can be combined into a balancing-free square-root algorithm. The chapter states a state-space characterization of all solutions to optimal Hankel-norm approximation and then describes an algorithm to compute an optimal Hankel-norm approximation.